{"id":"ac428269-a682-4c8c-8988-4e492608da3c","arxiv_id":"2501.08750","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Computes the monopole Floer homology and twisting involution of the complexity-2 protocork boundary, and constructs h-cobordisms of arbitrarily large Morgan-Szabó complexity between exotic pairs of closed 1-connected 4-manifolds.","lead":"This paper computes the Floer homology of the boundary of a standard protocork and uses it to build the first h-cobordisms of non-minimal, in fact arbitrarily large, complexity between closed exotic 4-manifolds. The same computation yields new strongly non-extendable cork involutions and refines the Morgan-Szabó complexity program.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The one-sentence b^+=1 extension in Remark 2 is the most load-bearing unverified input; it supports the arbitrarily-large-complexity part of Theorem 1.2.","rationale":"The paper's central new contribution is the computation of the monopole Floer homology of Y = dP_0 and the action of the twisting involution, together with the derivation of Theorem 1.2 from that computation. I read Sections 3-5 as the in-paper core; the computation is long and I cannot independently re-derive it here, but it is checked against published benchmarks such as the Akbulut cork and the first positron cork. The structural bridge from complexity to protocorks is imported from the unpublished [Lad22], so the reader's CONDITIONAL verdict is appropriate. I agree with the reader's identification of the imported classification as a weak spot, but I would rank the b^+=1 extension of [Lad22, Cor. 1.2] as the single most load-bearing concern: it is needed for every part of Theorem 1.2, whereas the uniqueness of the complexity-2 protocork is needed only for the specific m=17/N=2 assertion. The one-sentence Remark 2 is not a proof, and b^+=1 is precisely the regime where Seiberg-Witten invariants are metric dependent; a hidden wall-crossing term could destroy the bound. This is a verifiability concern, not an identified error. If the concrete test passes, the paper's main claim stands; if it fails, the arbitrary-large-complexity theorem loses its foundation. Since neither outcome is known, the appropriate verdict remains CONDITIONAL, unchanged from the reader.","tokens_in":20672,"tokens_out":7589,"duration_ms":71241,"concrete_test":"Obtain the full proof of [Lad22, Cor. 1.2] and trace every step that invokes b^+ > 1. Verify that the metric-dependent HM factorization [KM07, (3.14)] and the relevant exact triangles remain valid when b^+(M)=1, especially the step converting the U-torsion bound into a bound on formal dimension of a nonzero-variation moduli space. Independently compute the protocork twist relating K3 # overline{CP}^2 to 3CP^2 # 20 overline{CP}^2 (cited in Prop. 3.4): determine Delta, its U-torsion order l, and the maximal formal dimension d with nonzero mod-2 variation; if d > 2l, Remark 2 is false and Theorem 1.2(2) is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing unchecked input is Remark 2's one-sentence extension of [Lad22, Cor. 1.2] from b^+ > 1 to b^+ = 1. The constructed manifolds X_0 = E(1) and its blow-ups have b^+ = 1, so the contradiction in Theorem 6.3 needs the bound that no mod-2 Seiberg-Witten variation occurs for formal dimension > 2l, where l is the maximum U-torsion order of the difference element over the finite set of protocorks of complexity < r. If that bound fails in the b^+=1 case, the proof that complexity of C_m diverges, and hence items (1)-(2) of Theorem 1.2, collapses. Remark 2 asserts that the metric-dependent HM map and factorization formula [KM07, (3.14)] make the proof go through, but gives no argument that the wall-crossing and chamber issues specific to b^+=1 are controlled; in particular [KM07, pg. 562-566] treats the metric-dependent map, and its use in a b^+=1 gluing statement needs checking. The uniqueness of the complexity-2 protocork P_0 from [Lad22] is also load-bearing for the N=2 claim, but the b^+=1 extension is more central because it carries the arbitrarily-large-N part.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies Morgan-Szabó complexity 2 h-cobordisms in dimension 5. The author computes the monopole Floer homology \\widehat{HM}_*(Y) of the boundary Y of the protocork P0 and the action of the twisting involution τ on \\widehat{HM}_{-1}(Y) (Theorem 1.1). This computation is used to obstruct complexity-2 h-cobordisms and, via a Morgan–Szabó-style construction, to produce h-cobordisms from E(1)#17\\overline{CP}^2 to a Dolgachev or knot-surgered E(1)#17\\overline{CP}^2 of complexity greater than 2, and more generally of arbitrarily large complexity after sufficiently many blow-ups (Theorem 1.2). The paper also proves a strong-cork non-extendability criterion (Corollary 1.3). The proof of Theorem 1.1 combines surgery exact triangles, a computation of HF^+(Y_{-1,1,8}) via (1,1)-knot mapping cones, and published computations of the Akbulut cork boundary; the proof of Theorem 1.2 imports a protocork classification and a Seiberg-Witten variation bound from the author's unpublished preprint [Lad22].","tokens_in":20765,"tokens_out":9044,"duration_ms":88347,"significance":"If correct, Theorem 1.2 is a significant advance: it provides the first non-inertial h-cobordisms with non-minimal, in fact arbitrarily large, complexity between closed simply connected exotic 4-manifolds, and the factorial growth in the number of blow-ups is new. The Floer-theoretic computation of Theorem 1.1 is largely self-contained, and the reductions to published computations ([Gut24], [AD05], [LRS18]) are credible; Section 5's mapping-cone computation is detailed and checkable. The main caveat is that two load-bearing inputs come from the unpublished manuscript [Lad22], and the b^+=1 extension needed for the arbitrarily-large-complexity claim is asserted rather than proved. These issues are fixable in a revision but must be addressed before acceptance.","major_comments":[{"comment":"The divergent-complexity part of Theorem 1.2 depends on applying [Lad22, Cor. 1.2] to the manifolds X_{0,m}, which have b^+=1. Remark 2 states that the proof goes through using the metric-dependent HM map and the factorization formula [KM07, (3.14)], but it does not address the wall-crossing and chamber issues specific to b^+=1. Since equality (16) concerns chamber-dependent invariants m^\\pm and the contradiction step in Theorem 6.3 needs a bound on all Seiberg-Witten variations of formal dimension larger than 2ℓ, the b^+=1 case is load-bearing. A complete proof, or a precise published reference that covers b^+=1, must be supplied.","section":"Remark 2 / Theorem 6.3"},{"comment":"The assertion that P0 is the only complexity-2 protocork is imported from the unpublished preprint [Lad22]. This uniqueness is explicitly used to conclude that C_2 has complexity >2, and it also underlies the finiteness and the list S_r in (17). Please include the classification statement as a proved lemma in this paper or give a published reference; as it stands, the N=2 claim rests on an external input that the reader cannot verify.","section":"§6.1"},{"comment":"The proof of Lemma 6.2 uses the gluing formula [KM07, Prop. 27.5.1] for the invariants m_k and then asserts that m_k, together with k and the intersection pairing, determines the chamber-dependent invariants m^\\pm. For b^+=1 the invariants m^\\pm are metric- and chamber-dependent, and the manuscript does not justify this determination or the applicability of the gluing formula in that case. Since Lemma 6.2 is what produces the key equality (16), this step needs to be expanded rather than cited.","section":"Lemma 6.2"}],"minor_comments":[{"comment":"The sentence \"Since (τ_{0,1,8})_* acts as id on HM_red(Y_{0,1,8}) by Proposition 4.2\" is misstated: Proposition 4.2 concerns Y_{8,1,8}. The intended argument is that the image of HM_red(Y_{0,1,8}) in HM_red(Y_{8,1,8}) is contained in the fixed subspace FΔ ⊕ Fα of (τ_{8,1,8})_*, so the statement should be reworded accordingly.","section":"Proof of Proposition 4.3"},{"comment":"There are several typos, including \"occour\" in §6.1 and \"contraddiction\" in §6.3; the paper would benefit from a careful proofreading pass.","section":"§6.1 and §6.3"},{"comment":"The deduction of the displayed matrix for the action of (τ_{8,1,8})_* from the Lefschetz number 2 and the mod-2 Seiberg-Witten variation is compressed into the phrase \"as in [LRS18]\"; since this input is used for Theorem 1.1, the author should spell out the argument or cite the exact statement in [LRS18] that covers it.","section":"Proposition 4.2"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own unpublished manuscript [Lad22] for three load-bearing ingredients: the protocork classification, the statement τ*(x0)=x0+Δ, and Corollary 1.2. In addition, the b^+=1 extension in Remark 2 is a single sentence and could easily be missed by readers; I recommend requiring that this material be either proved in full or replaced by a published reference before acceptance. This is a scope and verifiability concern as much as a mathematical one."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper is worth reading: it gives the first construction of h-cobordisms of arbitrarily large Morgan-Szabó complexity between exotic pairs of closed, simply connected 4-manifolds, and it does so with a detailed, checkable computation of the monopole Floer homology of the protocork boundary Y=∂P0 and the action of the twisting involution. Theorem 1.2 genuinely extends [MS99] from inertial to non-inertial h-cobordisms, and the explicit statement that m_N=17 works for N=2, together with the factorial count of h-cobordisms, is concrete and new. The Floer-theoretic work in Sections 3–5 (surgery exact triangles, (1,1)-knot complexes, mapping cone formula) is laid out in enough detail that a patient reader can verify it, and the reductions to published computations by Guth, Akbulut–Durusoy, and Lin–Ruberman–Saveliev look sensible. Corollary 1.3 on strong non-extendability is a clean application.\n\nNow the soft spots. The construction leans on the author's unpublished preprint [Lad22] in two load-bearing places. First, the uniqueness of the complexity-2 protocork P0, imported from [Lad22], is essential for the N=2 claim; if a second complexity-2 protocork existed, the Theorem 1.1 obstruction would not rule out complexity-2 h-cobordisms between the constructed pairs. Second, and more central, the arbitrarily-large-N part of Theorem 1.2 rests on Remark 2's one-sentence extension of [Lad22, Cor. 1.2] from b^+>1 to b^+=1, via the metric-dependent HM-map and the factorization formula [KM07, (3.14)]. The stress-test note is right to flag this: the b^+=1 case has wall-crossing and chamber dependence that are not discussed, and this extension carries the divergence of complexity. That is a genuine gap in support, not a fatal flaw, but it needs to be checked carefully. There is also a garbled sentence in the proof of Theorem 6.3, and the grading assumptions AG are asserted rather than fully verified.\n\nOverall, the central construction is structurally sound and the in-paper computations are detailed enough to be checked. The paper is for people working on h-cobordisms, corks, and Floer-theoretic obstructions in 4-manifold topology. It deserves a serious referee, with explicit instructions to verify the b^+=1 extension and the [Lad22] imports.\n\nMy recommendation: send it to peer review.","headline":"Genuine new result with detailed Floer computations, but load-bearing reliance on unpublished [Lad22] and a one-sentence b^+=1 extension mean the referee should check those two spots carefully.","tokens_in":21527,"tokens_out":2408,"would_cite":true,"duration_ms":23102,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K40","57R58","57R57"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that h-cobordisms between exotic closed simply connected 4-manifolds can have arbitrarily large Morgan–Szabó complexity, with factorially many examples obtained from reflections of the intersection form.","keywords":["h-cobordism","Morgan–Szabó complexity","protocork","monopole Floer homology","Seiberg–Witten invariants","corks","exotic 4-manifolds","Dolgachev surfaces"],"falsifier":"Find a normal handle decomposition of the h-cobordism $C_2$ (17 blow-ups) with only two excess intersections, or independently compute $\\widehat{HM}^{-1}(\\partial P_0)$ and show that the action of $\\tau$ is not the unipotent matrix of Theorem 1.1 (for example, that $\\Delta = 0$ or that $\\tau_*$ is diagonalizable); either observation would break the obstruction and contradict Theorem 1.2(1).","tokens_in":20267,"feed_emoji":"🔄","tokens_out":15715,"duration_ms":129310,"temperature":0.7,"pith_summary":"This paper aims to show that the Morgan–Szabó complexity of an h-cobordism between two exotic closed simply connected 4-manifolds can be arbitrarily large, not merely for a manifold to itself but between genuinely distinct manifolds. The key step is a complete computation of the monopole Floer homology of the boundary of the protocork $P_0$ and of the action of the twisting involution $\\tau$, which yields an obstruction: a complexity-2 h-cobordism can only change Seiberg–Witten invariants in formal dimensions up to 2. Using this, the author constructs, via isometries of the intersection form, many h-cobordisms whose Seiberg–Witten variation has formal dimension $m^2 + m - 2$ diverging as $m \\to \\infty$, and shows that at least $2^{m_N} m_N!$ distinct examples exist for each $N$. In particular, blowing up the elliptic surface $E(1)$ and a Dolgachev or knot-surgery partner by 17 copies of $\\overline{\\mathrm{CP}}^2$ already produces h-cobordisms of complexity strictly greater than 2 (hence at least 4).","feed_headline":"Exotic pairs admit h-cobordisms of arbitrarily large complexity","feed_subtitle":"A Floer-theoretic computation rules out the minimal twist; 17 blow-ups already force complexity above 2.","key_machinery":"The load-bearing object is the protocork $P_0$: a compact 4-manifold with boundary $Y = \\partial P_0$ whose 'twist' operation — remove $P_0$ from a 4-manifold and reglue it via the boundary involution $\\tau$ — changes the smooth structure while preserving the homeomorphism type. $P_0$ is the unique protocork realizable at Morgan–Szabó complexity 2, and its boundary $Y$ is a graph manifold with plumbing graph given by two trivial normal bundle spheres intersecting three times with signs $+,-,+$. The computation of the monopole Floer homology $\\widehat{HM}_\\bullet(Y)$ and the action of $\\tau_*$ on $\\widehat{HM}^{-1}(Y)$ is the engine of the paper; it shows that the 'difference element' $\\Delta$ is $U$-torsion ($U\\cdot \\Delta = 0$), which bounds the variation of Seiberg–Witten invariants under the twist by formal dimension 2. The construction of the h-cobordisms then uses Kreck's theorem (isometry classes of intersection forms parametrize h-cobordisms), composing a homeomorphism-induced isometry $\\Phi$ with a reflection $\\rho_m$ along a class $\\alpha_m = (2m+1)H - \\sum E_i$ of square $-1$; Lemma 6.2 guarantees the isometry preserves chamber-dependent Seiberg–Witten invariants, so the variation along the resulting h-cobordism is exactly controlled by the reflection formula.","core_discovery":"The paper's central discovery is that the minimal non-trivial Morgan–Szabó complexity, 2, is obstructed by monopole Floer homology: for the unique complexity-2 protocork $P_0$, with boundary $Y = \\partial P_0$ and twist involution $\\tau$, the reduced Floer homology is $\\widehat{HM}^{-1}(Y) \\cong \\mathbb{F} x_0 \\oplus \\mathbb{F} \\Delta \\oplus \\mathbb{F} \\alpha_1$ with $\\tau_*(x_0) = x_0 + \\Delta$ and $U\\cdot \\Delta = 0$. Consequently a protocork twist using $P_0$ can alter Seiberg–Witten invariants only in formal dimension at most 2. The theorem that follows, Theorem 1.2, asserts that for $X_0 = E(1)$ and $X_1$ a Dolgachev surface or the Fintushel–Stern knot surgery on $E(1)$, blowing up to $X_i \\# 17\\overline{\\mathrm{CP}}^2$ yields at least $2^{17}17!$ h-cobordisms of complexity greater than 2, and for every $N>0$ there is some number $m_N$ of blow-ups giving at least $2^{m_N} m_N!$ h-cobordisms of complexity larger than $N$. The same Floer-theoretic input, via Corollary 1.3, yields strongly non-extendable cork involutions for the Akbulut cork and the $(Y_{1,8,1}, \\tau_{1,8,1})$ cork.","pith_inferences":["The factorial growth in the count of h-cobordisms likely reflects many genuinely distinct 5-dimensional h-cobordisms, not merely different isometries; distinguishing them would require 5-dimensional invariants, which the paper does not compute.","The mechanism is not special to $E(1)$: any exotic pair homeomorphic via an isometry that preserves chamber-dependent Seiberg–Witten invariants should admit arbitrarily complex h-cobordisms after enough blow-ups along similar reflection classes.","A testable consequence: an explicit normal handle decomposition of one of the constructed h-cobordisms, if it could be found, would either confirm the complexity lower bound or expose a gap in the Floer-theoretic obstruction.","The obstruction at complexity 2 is sharp in the sense that the moment $\\Delta$ becomes $U$-torsion of higher order, the allowed formal dimension grows; exploiting protocorks with larger torsion order might produce even stronger lower bounds with fewer blow-ups."],"forward_implications":["For every $N > 0$, some number $m_N$ of blow-ups of a fixed exotic pair yields at least $2^{m_N} m_N!$ distinct h-cobordisms of complexity larger than $N$, so complexity is unbounded among non-inertial h-cobordisms.","With 17 blow-ups, the construction already rules out the minimal non-trivial complexity: the h-cobordisms have complexity at least 4.","The same Floer-theoretic calculation gives a general criterion for strong non-extendability of cork involutions, and the Akbulut cork and the $(Y_{1,8,1}, \\tau_{1,8,1})$ cork satisfy it.","The computation of $\\widehat{HM}_\\bullet(Y)$ and of $\\tau_*$ covers a cyclic graph manifold case not treated by existing general algorithms for graph manifolds."],"supporting_citations":[{"why":"Supplies the classification of complexity-2 protocorks (only $P_0$), the difference element $\\Delta$, and the bound on Seiberg–Witten variation via $U$-torsion order used in the contradiction step.","marker":"[Lad22]"},{"why":"Provides monopole Floer homology theory, the gluing formula used in Lemma 6.2, and the relative Seiberg–Witten invariant conventions.","marker":"[KM07]"},{"why":"Defines Morgan–Szabó complexity and gives the inertial examples of diverging complexity that Theorem 1.2 extends to non-inertial pairs.","marker":"[MS99]"},{"why":"Establishes the bijection between h-cobordisms and isometries of intersection forms, used to construct the h-cobordisms $C_m$.","marker":"[Kre01]"},{"why":"Supplies the computation of the Akbulut cork involution action and the Lefschetz-number argument used in Proposition 4.2 and Corollary 1.3.","marker":"[LRS18]"},{"why":"Computes the Heegaard Floer homology of the first positron cork boundary with $\\mathbb{F}$ coefficients, used as input for Proposition 3.3.","marker":"[Gut24]"},{"why":"Provides the mapping cone formula used in Section 5 to compute $\\widehat{HF}^+(Y_{-1,1,8})$ from the $(1,1)$-knot surgery.","marker":"[OS08]"},{"why":"Defines the knot surgery construction producing the exotic partner $X_1$ of $E(1)$.","marker":"[FS98]"}],"fun_headline_variants":["Floer homology blocks h-cobordism of complexity 2","Blow-ups give h-cobordisms of any complexity","Seventeen blow-ups force non-minimal h-cobordisms","Arbitrarily complex h-cobordisms from exotic pairs","Monopole Floer theory rules out minimal protocork twist"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument relies on the claim that at the lowest non-zero complexity there is only one protocork shape, $P_0$, a classification imported from the author's previous work; if a second shape with the same complexity existed, the obstruction in Theorem 1.1 would not rule out complexity-2 h-cobordisms between the constructed pairs.","fun_headline_variants_meta":{"raw":{"variants":["Floer homology blocks h-cobordism of complexity 2","Blow-ups give h-cobordisms of any complexity","Seventeen blow-ups force non-minimal h-cobordisms","Arbitrarily complex h-cobordisms from exotic pairs","Monopole Floer theory rules out minimal protocork twist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1635,"prompt_tokens":998,"completion_tokens":637,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":544}},"tokens_in":614,"tokens_out":637,"duration_ms":5655,"temperature":1.0,"reasoning_tokens":544,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:21:17.907262+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a normal handle decomposition of the h-cobordism $C_2$ (17 blow-ups) with only two excess intersections, or independently compute $\\widehat{HM}^{-1}(\\partial P_0)$ and show that the action of $\\tau$ is not the unipotent matrix of Theorem 1.1 (for example, that $\\Delta = 0$ or that $\\tau_*$ is diagonalizable); either observation would break the obstruction and contradict Theorem 1.2(1).","supporting_citations":[],"review_version":1}